Many of the results you mention form part of the categorical approach to (parts of) Universal Algebra. There is a lot of active research in this area, especially in the study of semi-abelian categories, Malt'sev categories, etc. Although is not main stream Universal algebra and some universal algebraists may not feel it deserves mention as part of that subject, it may help to look at the book by Bourn and Borceux mentioned in the n-Lab article on semi-abelian categories.

Many of the results you mention form part of the categorical approach to (parts of) Universal Algebra. There is a lot of active research in this area, especially in the study of semi-abelian categories, Malt'sev categories, etc. Although this is not main stream Universal algebra and some universal algebraists may not feel it deserves mention as part of that subject, it may help to look at the book by Bourn and Borceux mentioned in the n-Lab article on semi-abelian categories.